One last idea to finish up the one coordinate case is to consider the inclusion of matter fields and how they interact. While obviously quite simple with just one coordinate, we can show how the scheme could be extended to higher numbers of property coordinates. The first type we will consider are scalar fields, adhering to our previous rules regarding spin-statistics the superscalar field Φ(X) is overall Bose and thus must consist of terms containing an even number of property coordinates. Thus it must take the general form:
Φ(x, ζ,ζ¯) =U(x) +V(x) ¯ζζ. (4.57)
Imposing selfduality on this we find that the field can be reduced to the form:
Φ(X) =ϕ(x)(1 + ¯ζζ)/2. (4.58)
φcarries zero charge and could be thought of as an analogue to the Higgs field, though with only one property coordinate we lack the ability to properly identify it. We now couple this to our metric, though we will drop the ci curvature to simplify the results. A mass term in the Lagrangian of µ2ϕ2/2 will arise through the property integral:
(`2/2)
Z
dζdζ¯√−G.. µ2Φ2=
Z
4.12. SUMMARY 55
The kinetic term is a bit more involved as the ζ and ¯ζ derivatives contribute to the mass as well, thus we consider:
(l2/2)
Z
dζdζ¯√−G.. GM N∂NΦ∂MΦ. (4.60)
Upon including the metric from 4.13 we find the contributions from the gauge field cancel out as required and we are left with:
Z
dζdζ¯√−g..[(1 + 2 ¯ζζ)gmn∂nϕ ∂mϕ/4 + ¯ζζϕ2/l2]. (4.61) Thus the only way to obtain a massless scalar field is to match the kinetic mass term ϕ2/l2
from Equation 4.61 with the previously constructed mass term in Equation 4.59.
Moving on to spinor fields we now need to generalise the Dirac equation to our graded superspace. The natural way to do this is by taking iγaeam∂m to iΓAEAM∂M, the trick is now to determine the extended Dirac matrices ΓM. If the Dirac operator acts on a spino- rial superfield of the form Ψ(X) = θζψ¯ (x), then the following representation for the Dirac matrices works:
Γa=γa, lΓζ = 2i∂/∂θ, lΓζ¯= 2iθ, (4.62)
where θis another complex anti-commuting scalar that we eventually need to integrate over
θ and ¯θ. The action of the extended Dirac operator then yields:
iΓAEAM∂MΨ = iγaeam∂m+eγaAaζ¯ ∂ ∂ζ¯+ 2 l(1−fζζ¯ ) ∂2 dθdζ¯ θζψ¯ = θζγ¯ aeam(i∂m+eAm)ψ− 2 l(1−fζζ¯ )ψ. (4.63)
When we include the adjoint ¯Ψ≡ −ψζ¯ θ¯and integrate over ζ,ζ, θ¯ and ¯θwe end up with the normal gauge invariant spinorial Lagrangian density:
L = Z (dζdζ¯)(dθdθ¯) ¯Ψ(X)iΓAEAM∂M − M Ψ(X) = ψ¯(x) [γaeam(i∂m+eAm)− M]ψ(x). (4.64) While everything works out nicely, this is obviously a very simple system. The representation of the extended Dirac matrices ΓM will need to be revisited in order to include chirality, if we are to encompass all spin states.
4.12
Summary
In this chapter we have produced a super metric with one complex property coordinate. To enforce that it transforms correctly as a rank 2 covariant tensor under locally varying space time transformations we introduce a gauge field; these transformations are then none other
than gauge transformations. We generalise the metric by including factors that act like cur- vature in property space and then calculate Christoffel symbols, Ricci tensor components and the Ricci super scalar. The result of this is a Einstein-Hilbert Lagrangian density that unifies gravity with electromagnetism, as well as producing a cosmological constant. This cosmo- logical constant has the wrong sign, but this may well be remedied by additional property coordinates. We also consider variation of the Lagrangian density with respect to the metric and gauge field, to get Maxwell equations and field equations consistent with the Einstein- Hilbert Lagrangian. Finally we briefly consider matter fields, both scalar and spinor and find they work out satisfactorily. The next step is to consider a more involved model, with two property coordinates.
The work done in this chapter appears in Delbourgo and Stack (2014), though in less detail than this chapter. The Mathematica code used to generate the results is available from the UTAS digital repository, and is documented in Appendix B.
Chapter 5
General Relativity with two
property coordinates
In this chapter we now consider a model with two complex coordinates, following the process from Chapter 4. We introduce an SU(2) gauge field to our 4+4 dimensional metric and calculate the resulting Ricci tensor, Ricci scalar and field equations. The result is gravity unified with SU(2) Yang-Mills, the non-abelian version of the result of the last chapter. Without including chirality however we cannot model the weak force, so rather this serves to demonstrate that our model can be extended to multiple property coordinates.
5.1
Notation
Introducing multiple property coordinates results in having to deal with property indices, we adopt the labelling from Chapter 3. Upper case Roman letters like M, N, Lrepresent indices that run across both graded even and graded odd elements, lower case Roman letters like
m, n, l represent space-time indices and Greek lettersµ, ν, λrepresent odd graded indices. It is also useful to note that contravariance and covariance do not apply to the property indices in the same way they do to space-time indices, we will elaborate on this shortly.
5.2
Extended Minkowski metric
Our starting point to building our metric is the following metric distance for a flat 4+4 dimensional graded manifold:
ds2 =dXMdXNIN M =dxmdxnηnm+ 1 2l 2dζµdζ¯νη ¯ νµ+ 1 2l 2dζµ¯dζνη νµ¯. (5.1) 57
This results in the extended Minkowski metric taking the following form: IN M = ηnm 0 0 0 0 12l2ηνµ¯ 0 12l2ηνµ¯ 0 , (5.2)
where ηµ¯ν =δµν¯,ηνµ¯ =δνµ¯ and ηµν¯=−ηνµ¯ . Note that I is graded symmetric,
IM N = (−1)M NIN M. (5.3) The space-time piece,ηnm is simply the Minkowski metric, and is used to swap between con- travariant and covariant indices in flat space-time. The property sector piece, ηνµ¯ is used to swap between raised and lowered property indices. As the coordinates themselves are scalar, there is no dependence on the curvature of space-time and hence the raising and lowering of property indices using ηµν¯ and its inverse ηµ¯ν = δµ¯ν can be performed even in curved space-time.
Like in the one coordinate case, this flat-space metric is not invariant under space-time dependent phase transformations in property. Consider a transformation of the form:
xm →xm, ζµ→eiΘ(x) µν¯ ζν, ζµ¯ →ζν¯ e−iΘ(x) νµ¯ . (5.4)
We again require our metric to transform as a rank 2 covariant tensor, but if we consider the following:
IM N = (−1)R(S+N)∂M0R∂N0SI0RS, (5.5) and look at the space-time component under the property phase transformation we find:
Imn=(−1)RS∂m0R∂n0SI0RS, =∂m0r∂n0sI0rs− 1 2∂m 0ρ∂ n0¯σI0ρσ¯− 1 2∂m 0¯ρ∂ n0σI0ρσ¯ , (5.6) =ηmn− 1 2l 2ζµ¯∂ ne−iΘ(x) µσ¯ δσρ¯ ∂meiΘ(x) ρν¯ ζν − 1 2l 2ζν¯∂ me−iΘ(x) νρ¯ δρσ¯ ∂neiΘ(x) σµ¯ ζµ.
The last two terms do not cancel, similar to the abelian case, and soIM N does not transform correctly as a rank 2 covariant tensor. Thus to produce our true metric GM N we need to introduce a non-abelian gauge field Wmµ¯ν. Before we do this however we need to discuss the notation for matrices in the property sector.